On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L\sp 2$-normalized eigenfunctions $\{\phi\sb \lambda\}$ satisfy $\lvert \rvert\phi\sb \lambda\lvert \rvert\sb\infty\leq C\lambda\sp {(n-1)/2}$, where $-\Delta\phi\sb \lambda=\lambda\sp 2\phi\sb \lambda$. The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S\sp n$. But, of course, it is not sharp for many Riemannian manifolds, for example, for flat tori $\mathbb {R}\sp n/\Gamma$. We say that $S\sp n$, but not $\mathbb {R}\sp n/\Gamma$, is a Riemannian manifold with maximal eigenfunction growth. The problem that motivates this paper is to determine the $(M, g)$ with maximal eigenfunction growth. Our main result is that such an $(M, g)$ must have a point $x$ where the set $\mathscr {L}\sb x$ of geodesic loops at $x$ has positive measure in $S\sp \ast\sb xM$. We show that if $(M, g)$ is real analytic, this puts topological restrictions on $M$; for example, only $M=S\sp 2$ or $M=\mathbb {R}P\sp 2$ (topologically) in dimension $2$ can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any $M$ fail to have maximal eigenfunction growth. In addition, we construct an example of $(M, g)$ for which $\mathscr {L}\sb x$ has positive measure for an open set of $x$ but which does not have maximal eigenfunction growth; thus, it disproves a naive converse to the main result.

We say that a group is almost abelian if every commutator is central and squares to the identity. Now let $G$ be the Galois group of the algebraic closure of the field $\mathbb {Q}$ of rational numbers in the field $\mathbb {C}$ of complex numbers. Let $G\sp {ab+\epsilon}$ be the quotient of $G$ universal for continuous homomorphisms to almost abelian profinite groups, and let $\mathbb {Q}\sp {ab+\epsilon}/\mathbb {Q}$ be the corresponding Galois extension. We prove that $\mathbb {Q}\sp {ab+\epsilon}$ is generated by the roots of unity, the fourth roots of the rational primes, and the square roots of certain algebraic sine-monomials. The inspiration for the paper came from recent studies of algebraic $\Gamma$-monomials by P. Das and by S. Seo.

The intertwining operators $A\sb d=A\sb d(g)$ on the round sphere $(S\sp n,g)$ are the conformal analogues of the power Laplacians $\Delta\sp {d/2}$ on the flat $\mathbf {R}\sp n$. To each metric $\rho g$, conformally equivalent to $g$, we can naturally associate an operator $A\sb d(\rho g)$, which is compact, elliptic, pseudodifferential of order $d$, and which has eigenvalues $\lambda\sb j(\rho)$; the special case $d=2$ gives precisely the conformal Laplacian in the metric $\rho g$. In this paper we derive sharp inequalities for a class of trace functionals associated to such operators, including their zeta function $\sum\sp j\lambda\sp j(\rho)\sp {-s}$, and its regularization between the first two poles. These inequalities are expressed analytically as sharp, conformally invariant Sobolev-type (or log Sobolev type) inequalities that involve either multilinear integrals or functional integrals with respect to $d$-symmetric stable processes. New strict rearrangement inequalities are derived for a general class of path integrals.

The maximal function along a curve $(t,\gamma(t),t\gamma(t))$ on the Heisenberg group is discussed. The $L\sp p$-boundedness of this operator is shown under the doubling condition of $\gamma\sp \prime$ for convex $\gamma$ in $\mathbb {R}\sp +$. This condition also applies to the singular integrals when $\gamma$ is extended as an even or odd function. The proof is based on angular Littlewood-Paley decompositions in the Heisenberg group.